XJTLU Entrepreneur College (Taicang) Cover Sheet
Module code and Title DTS203TC Design and Analysis of Algorithms
School Title School of AI and Advanced Computing
Assignment Title Coursework
Submission Deadline Sunday, March 24th 23:59 (UTC+8 Beijing), 2024
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DTS203TC Design and Analysis of Algorithms
Coursework
Deadline: Sunday, March 24th 23:59 (UTC+8 Beijing), 2024
Percentage in final mark: 40%
Learning outcomes assessed:
A. Describe the different classes of algorithms and design principles associated with them;
Illustrate these classes by examples from classical algorithmic areas, current research and
applications.
B. Identify the design principles used in a given algorithm, and apply design principles to produce
efficient algorithmic solutions to a given problem.
C. Have fluency in using basic data structures in conjunction with classical algorithmic problems.
Late policy: 5% of the total marks available for the assessment shall be deducted from the
assessment mark for each working day after the submission date, up to a maximum of five working
days
Risks:
• Please read the coursework instructions and requirements carefully. Not following these
instructions and requirements may result in loss of marks.
• The assignment must be submitted via Learning Mall to the correct drop box. Only electronic
submission is accepted and no hard copy submission.
• All students must download their file and check that it is viewable after submission.
Documents may become corrupted during the uploading process (e.g. due to slow internet
connections). However, students themselves are responsible for submitting a functional and
correct file for assessments.
• Academic Integrity Policy is strictly followed.
• The use of Generative AI for content generation is not permitted on this assessed coursework.
Submissions will be checked through Turnitin.
Overview
In this coursework, you are expected to design and implement algorithms to produce solutions to
four given problems (Tasks **4) in Python. For Tasks **4, you should have function(s) to receive
task input as parameters, implement your algorithm design and return results. You also need to
write a short report answering a list of questions in Task 5 that are related to the given four
problems.
Task 1 (15 marks)
You have n coins and a balance. Among the coins, there is one fake coin which has different weight
than the other coins. All the coins, except the fake coin, have exactly the same weight. Utilizing
the balance, you can compare the weight of any number of coins on each pan. The balance will
indicate whether the set of coins on one pan weighs the same as the set on the other pan. Assume
the number of coins is a power of 3 (n = 3k
). Implement an efficient algorithm to find the fake coin.
Input: coins = [10, 10, 10, 9, 10, 10, 10, 10, 10]
Output: 4
Explanation: In the given example, there are 9 coins, with the genuine coins weighing 10 units
each and the fake coin weighing 9 units. The fourth (output) coin is identified as the fake coin.
You should have a function named findFakeCoin to receive a list of integers and return the index
of fake coin (int). Please consider the time complexity when you design your algorithm. A naïve
approach will result in loss of marks.
Task 2 (15 marks)
Consider a d-ary heap as a generalization of the binary heap, where each node has d children
instead of 2. Implement a d-ary max heap and performs heap sort on a given array.
Example:
Input: nums = [7, 6, 5, 9, 8], d = 3
Output: [5, 6, 7, 8, 9]
You should create a function named dHeapSort that takes a list of integers to be sorted and an
integer d indicates d-ary as parameters. The function should return a list containing the sorted
integers based on the d-ary heap sort algorithm.
Task 3 (15 marks)
You’re driving an electric car from Shanghai to Beijing, where there are charging stations along
the way at distances x1, x2, …, xn from Shanghai. Due to varying wait times c and charge speeds g,
charging k kilometers worth of electric at station xi takes ci +kgi minutes. Your car has a capacity
sufficient to travel 400 kilometers on a single charge. Assume car battery begin with 0 at the first
station x1 in Shanghai and xn is your destination in Beijing. Design an efficient algorithm that finds
where you should stop to spend the minimum amount of time at charging stations during your trip.
Example:
Input: x = [0, 100, 300, 600, 800, 1000], c = [0, 10, 0, 20, 10, 0], g = [0.05, 0.2, 0.1, 0.2, 0.1, 0]
Output: [20,0,30,40,30,0]
Explanation: In the provided example, there are a total of 6 stations. The objective is to drive
from the first station to the final one, optimizing the time spent at each station. The output details
the time spent at each station, aiming to minimize the overall time during the trip. For instance,
the time spent at the first station is calculated as 0 + 400 * 0.05 = 20 minutes.
You should have a function named timeSpent to receive the information of distance x (List[int]),
wait time c (List[int]), charge speed g (List[int]) and return the time spend at each station t
(List[int]). Please consider the time complexity when you design your algorithm. A naïve
approach will result in loss of marks.
Task 4 (15 marks)
Consider an undirected graph G = (V, E) with distinct weights assigned to each edge. Your task is
to find the minimum spanning tree (MST), denoted as T, on G. Anticipating the potential removal
of one edge in the future, implement an efficient algorithm to compute the MST T along with a
backup edge, denoted as r(e), for each edge e in T. This ensures that if any edge e is removed,
adding its corresponding backup edge r(e) to the tree will quickly create a new minimum spanning
tree in the modified graph.
Example:
Input: graph = {'A': {'B': 4, 'C': 8},
'B': {'A': 4, 'C': 2, 'D': 3, 'E': 5},
'C': {'A': 8, 'B': 2, 'D': 1, 'E': 6},
'D': {'B': 3, 'C': 1, 'E': 7},
'E': {'B': 5, 'C': 6, 'D': 7}}
Output: {('A', 'B'): ('A', 'C'), ('B', 'C'): ('B', 'D'), ('C', 'D'): ('B', 'D'), ('B', 'E'): ('C', 'E')}
Explanation: The input graph (as shown in figure) can be represented as a dictionary where the
keys are vertices, and the values are dictionaries representing the edges going out of that vertex
and the weights of those edges. The output can also be represented as a dictionary where the keys
are the edges of the minimum spanning tree T, and the values are the corresponding backup edge
of each edge in T. For example, if edge ('A', 'B') is removed from the graph G, adding backup edge
('A', 'C') will form a new minimum spanning tree.
You should have a function named modifiedMST to receive the graph represented as a dictionary
and return the minimum spanning tree T with backup edges represented as a dictionary. Please
consider the time complexity when you design your algorithm. A naïve approach will result in
loss of marks.
Task 5 (40 marks)
Answer the following questions in your report. Maximum 800 words for Task 5. (Clarity and
brevity are valued over length).
T5-1: For Task 1, give a recurrence of the running time of your algorithm and solve the recurrence
with two different methods.
T5-2: For Task 2, what is the running time of d-ary heap’s insert and heapify operations? Will dary heap sort faster than binary heap sort? Justify your answer.
T5-3: For Task 3, explain the design, show the correctness, and analyse the time and space
complexity of your algorithm.
T5-4: Given the same graph G from Task 4, is it possible to have more than one minimum spanning
tree T? Justify your answer.
Submission
Electronic submission on Learning Mall is mandatory. You need to submit a zip file (named
DTS203TC-CW-YOUR_NAME.zip) containing the following documents.
1. Cover letter with your student ID.
2. Your source code for Tasks **4: Solutions.ipynb
3. A pdf file contains all the source code (should be the same as the submitted ipynb file)
and your report (task 5). You can also write the report in jupyter notebook and export as a
pdf file.
Generic Marking Criteria
Grade Point
Scale
Criteria to be satisfied
A 81+ First ➢ Outstanding work that is at the upper limit of
performance.
➢ Work would be worthy of dissemination under
appropriate conditions.
➢ Mastery of advanced methods and techniques at a
level beyond that explicitly taught.
➢ Ability to synthesise and employ in an original way
ideas from across the subject.
➢ In group work, there is evidence of an outstanding
individual contribution.
➢ Excellent presentation.
➢ Outstanding command of critical analysis and
judgment.
B 70 - 80 First ➢ Excellent range and depth of attainment of intended
learning outcomes.
➢ Mastery of a wide range of methods and techniques.
➢ Evidence of study and originality clearly beyond the
bounds of what has been taught.
➢ In group work, there is evidence of an excellent
individual contribution.
➢ Excellent presentation.
➢ Able to display a command of critical thinking,
analysis and judgment.
C 60 - 69 Upper
Second
➢ Attained all the intended learning outcomes for a
module or assessment.
➢ Able to use well a range of methods and techniques
to come to conclusions.
➢ Evidence of study, comprehension, and synthesis
beyond the bounds of what has been explicitly
taught.
➢ Very good presentation of material.
➢ Able to employ critical analysis and judgement.
➢ Where group work is involved there is evidence of a
productive individual contribution
D 50- 59 Lower
Second
➢ Some limitations in attainment of learning
objectives but has managed to grasp most of them.
➢ Able to use most of the methods and techniques
taught.
➢ Evidence of study and comprehension of what has
been taught
➢ Adequate presentation of material.
➢ Some grasp of issues and concepts underlying the
techniques and material taught.
➢ Where group work is involved there is evidence of a
positive individual contribution.
E 40 - 49 Third ➢ Limited attainment of intended learning outcomes.
➢ Able to use a proportion of the basic methods and
techniques taught.
➢ Evidence of study and comprehension of what has
been taught, but grasp insecure.
➢ Poorly presented.
➢ Some grasp of the issues and concepts underlying
the techniques and material taught, but weak and
incomplete.
F 0 - 39 Fail ➢ Attainment of only a minority of the learning
outcomes.
➢ Able to demonstrate a clear but limited use of some
of the basic methods and techniques taught.
➢ Weak and incomplete grasp of what has been
taught.
➢ Deficient understanding of the issues and concepts
underlying the techniques and material taught.
➢ Attainment of nearly all the intended learning
outcomes deficient.
➢ Lack of ability to use at all or the right methods and
techniques taught.
➢ Inadequately and incoherently presented.
➢ Wholly deficient grasp of what has been taught.
➢ Lack of understanding of the issues and concepts
underlying the techniques and material taught.
➢ Incoherence in presentation of information that
hinders understanding.
G 0 Fail ➢ No significant assessable material, absent, or
assessment missing a "must pass" component.
Marking Criteria
Tasks 100 Components Description Maximum
Credit Mark
Task 1 15
Implementation
6 marks
Correct function definition [0/1 mark] 1
Correct algorithm design [0/2 marks] 2
Algorithm implementation [0-3 marks] 3
Evaluation
8 marks
Time complexity [0/3 marks] 3
5 test cases will be used to evaluate the
correctness of the function. 1 mark for each
test case.
5
Code quality
1 mark Readability, Formatting, Comments 1
Task 2 15
Implementation
9 marks
Correct function definition [0/1 mark] 1
Data structure implementation [0/2/4
marks] 4
Algorithm implementation [0/2/4 marks] 4
Evaluation
5 marks
5 test cases will be used to evaluate the
correctness of the function. 1 mark for each
test case.
5
Code quality
1 mark Readability, Formatting, Comments 1
Task 3 15 Implementation Correct function definition [0/1 mark] 1
6 marks Correct algorithm design [0/2 marks] 2
Algorithm implementation [0-3 marks] 3
Evaluation
8 marks
Time complexity [0/3 marks] 3
5 test cases will be used to evaluate the
correctness of the function. 1 mark for each
test case.
5
Code quality
1 mark Readability, Formatting, Comments 1
Task 4 15
Implementation
6 marks
Correct function definition [0/1 mark] 1
Correct algorithm design [0/2 marks] 2
Algorithm implementation [0-3 marks] 3
Evaluation
8 marks
Time complexity [0/3 marks] 3
5 test cases will be used to evaluate the
correctness of the function. 1 mark for each
test case.
5
Code quality
1 mark Readability, Formatting, Comments 1
Task 5 40
Task 5-1
9 marks
Recurrence of the algorithm is correct [0/3
marks] 3
Solve recurrence with method 1 [0/3 marks] 3
Solve recurrence with method 2 [0/3 marks] 3
Task 5-2
9 marks
Running time of insert operation [0-3
marks] 3
Running time of heapify operation [0-3
marks] 3
Analysis of d-ary heap sort [0-3 marks] 3
Task 5-3
9 marks
Algorithm design [0/2 marks] 2
Correctness [0/3 marks] 3
Time and space complexity [0/2/4 marks] 4
Task 5-4
9 marks
‘Yes/No’ answer correct 3
Number of minimum spanning tree on G is
well justified [0-6 marks] 6
Report quality
4 marks
Fluency and readability [0/2 mark]
Formatting and conciseness [0/2 mark] 4
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